WASHINGTON — President Trump said on Saturday that the F.B.I. will have “free rein” to investigate allegations of sexual misconduct against Judge Brett M. Kavanaugh, but the emerging contours of the inquiry showed its limited scope.

Four witnesses will be questioned in coming days about aspects of the assault accusations against Judge Kavanaugh, according to two people familiar with the matter. Left off the list were former classmates who have contradicted Judge Kavanaugh’s congressional testimony about his drinking and partying as a student.

The White House will decide the breadth of the inquiry, though presidential advisers were working in concert with Senate Republicans, said the two people, one a senior administration official, who both spoke on the condition of anonymity to discuss a sensitive investigation.

The White House can order investigators to further examine the allegations if their findings from the four witness interviews open new avenues of inquiry, and Mr. Trump seemed to stress that part of the plan in a tweet late on Saturday.

“I want them to interview whoever they deem appropriate, at their discretion,” Mr. Trump wrote. He denied an NBC News report that he was limiting the inquiry and that investigators were not permitted to examine the claims of Julie Swetnick, a woman who has said she witnessed a severely drunken Judge Kavanaugh mistreat women at parties in high school, and that he had attended parties where high school boys gang-raped teenage girls.

Investigators will interview one of the witnesses, a high school friend of Judge Kavanaugh’s named Mark Judge, about Ms. Swetnick’s accusations, the two people said.

The inquiry, which will last no more than a week, is a limited background check of Judge Kavanaugh, not a full-fledged criminal investigation. He has vigorously denied any sexual impropriety or wrongdoing.

Democrats, left out of the discussions that led to Mr. Trump’s order, tried on Saturday to clarify the scope of the F.B.I. investigation with Senate Republicans and the White House. Senate Republicans drafted the witness list for the background check, according to the people familiar with it, and the Senate majority leader, Mitch McConnell, shared it with the White House.

The background check is the latest twist in a drama that has gripped the Capitol for the past two weeks, since a Northern California research psychologist named Christine Blasey Ford alleged that a young Mr. Kavanaugh attempted to rape her at a house party when they were teenagers. Mr. McConnell has said senators will vote on whether to confirm Judge Kavanaugh after the F.B.I. concludes its work.

The bureau will investigate aspects of the allegations made by Dr. Blasey, Ms. Swetnick and a third woman, Deborah Ramirez. She has said Judge Kavanaugh exposed himself to her during their freshman year at Yale. Ms. Ramirez’s lawyer, John Clune, said on Saturday that the F.B.I. had “reached out to interview Ms. Ramirez, and she has agreed to cooperate with their investigation.”

The F.B.I. plans to ask Mr. Judge about both Ms. Swetnick’s and Dr. Blasey’s allegations. Ms. Swetnick has said that Mr. Judge was at the parties where she saw Judge Kavanaugh mistreat women, and Dr. Blasey has said he was in the room when Judge Kavanaugh assaulted her. He has denied the allegations of both women.

The F.B.I. did not plan to question Ms. Swetnick herself, according to the two people familiar with the matter. Ms. Swetnick’s lawyer, Michael Avenatti, said on Twitter on Saturday that they had not heard from the bureau.

The four witnesses were Mr. Judge; Leland Keyser, a high school friend of Dr. Blasey’s who she said attended the party but was not told of the assault; P.J. Smyth, another party guest; and Ms. Ramirez, the Yale accuser.

The White House has asked that the F.B.I. share its findings after investigators complete those interviews, and at that point, Mr. Trump and his advisers would decide whether to have the accusations investigated further, the people said.

Led by Donald F. McGahn II, the White House counsel, Mr. Trump’s advisers are helping direct the scope of the background check, according to the senior administration official. Mr. McGahn shared the witness list with the F.B.I. but is working in concert with Senate Republicans, and senators considered key swing votes have had extensive input, the people said. Senators Susan Collins of Maine and Jeff Flake of Arizona have both said they want Mr. Judge questioned by the F.B.I.

The witness list did not extend to high school and college classmates who have said in interviews that Judge Kavanaugh drank heavily, including some who said he went beyond typical consumption.

During Thursday’s hearing, he spoke of enjoying beer. “I drank beer with my friends,” he said. “Almost everyone did. Sometimes I had too many beers. Sometimes others did. I liked beer. I still like beer. But I did not drink beer to the point of blacking out,” he said.

A Yale classmate of his who said they frequently drank together, Lynne Brookes, said after the hearing that he had “grossly misrepresented and mischaracterized his drinking.” She was roommates with Ms. Ramirez.

Democrats raised concerns about the scope of the background check on Saturday.

“We have been concerned from the outset about the so-called limits on scope, not to mention time for this investigation,” said Senator Richard Blumenthal, Democrat of Connecticut and a member of the Judiciary Committee. “It has to be full fair, real, not check-the-box. So any limits should be viewed with serious question.”

Senator Kamala Harris, Democrat of California, called reports of the White House’s role in the inquiry “outrageous” and said on Twitter, “The White House should not limit or interfere with the FBI investigation in any way.”

Mr. Trump had kept unusually silent on the issue until a raucous rally Saturday night in Wheeling, W.Va., where he sought to tap into Republican anger about Democrats’ treatment of Judge Kavanaugh as a way to energize his party’s voters in the midterm elections.

He said that the “entire nation has witnessed the shameless conduct of the Democrat Party.” The crowd ate it up, cheering for Mr. Trump and booing loudly when he mentioned Democrats.

The president viciously mocked Senator Dianne Feinstein, Democrat of California, acting mentally confused as he pretended to be Ms. Feinstein responding to a question about whether her staff had leaked information about one of Judge Kavanaugh’s accusers.

“Remember her answer? Did you leak the document?” he said. “Uh, uh, what, no? I didn’t. Leak? Wait a minute.” He then pretended to be Ms. Feinstein, the top Democrat on the Judiciary Committee, turning back to look at a staff member. “No, we didn’t leak it.”

The crowd applauded boisterously, apparently approving of the president’s mocking of an 85-year-old senator. Later, he called Democrats “disgraceful political hacks” and accused them of trying to stop Judge Kavanaugh’s nomination because they are “nasty” and “angry people” who are willing to sacrifice anyone who gets in their way.

“You see the meanness, the nastiness,” Mr. Trump said. “They don’t care who they hurt, who they have to run over in order to get power and control. That’s what they want, power and control. We are not going to give it to them.”

Before leaving Washington for the rally, he told reporters that he expected the background check to go well. “They have free rein,” he said of investigators. “They can do whatever they have to do, whatever it is that they do.”

Mr. Trump is said to have privately made clear that he will hold Senate Republicans accountable if Judge Kavanaugh does not get through, according to one person familiar with his conversations in recent days.

At least one of the potential witnesses, Ms. Keyser, notified the Judiciary Committee on Saturday that she would cooperate with the F.B.I. investigation.

Ms. Keyser, a longtime friend of Dr. Blasey’s, has said she does not recall the gathering. But her lawyer, Howard J. Walsh III, said in a letter to the committee that her lack of a memory of the gathering does not mean she does not believe Dr. Blasey.

“Notably, Ms. Keyser does not refute Dr. Ford’s account, and she has already told the press that she believes Dr. Ford’s account,” her lawyer wrote. Dr. Blasey sometimes goes by her married name, Ford.

The delay in a full Senate vote on Judge Kavanaugh’s confirmation to accommodate the new examination was forced on Friday by Mr. Flake, Ms. Collins and Senator Lisa Murkowski of Alaska. Without their votes to confirm the judge, Mr. Trump and Republican leaders in the Senate had no choice but to request an F.B.I. inquiry — a move that Democrats had demanded for days.

Mr. Flake, who is retiring, described to The Atlantic his last-minute decision to change his mind on moving ahead with a quick confirmation of Judge Kavanaugh. Mr. Flake said he was motivated to call for a delay as a way to preserve the institutions of the Senate and the Supreme Court.

“The Supreme Court is the lone institution where most Americans still have some faith. And then the U.S. Senate as an institution — we’re coming apart at the seams,” Mr. Flake said, echoing remarks he made when he announced his retirement. “There’s no currency, no market for reaching across the aisle. It just makes it so difficult.”